🔬 mesoscale physics

Disorder-Induced Entanglement Phase Transitions in Non-Hermitian Systems with Skin Effects

This paper investigates disorder-induced entanglement phase transitions in non-Hermitian Hatano-Nelson models, revealing that weak disorder transforms the area-law entanglement scaling into a logarithmic regime before a critical disorder strength triggers a transition back to area-law behavior with universal algebraic scaling at the critical point.

Original authors: Kai Li, Ze-Chuan Liu, Yong Xu

Published 2026-08-05

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Kai Li, Ze-Chuan Liu, Yong Xu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where everyone is trying to move in a specific direction. In the world of quantum physics, particles are the dancers, and the rules of the dance are written by something called a "Hamiltonian." Usually, these rules are perfectly balanced, like a seesaw that never tips. But in recent years, scientists have become fascinated by "non-Hermitian" systems, where the rules are unbalanced. Imagine a dance floor where the floor itself is slightly tilted, or where the music pushes everyone to the left. In these systems, particles don't just spread out; they all get shoved to one edge of the room, piling up in a corner. This phenomenon is known as the "skin effect," and it changes how the particles interact with each other.

Now, add a third ingredient: "disorder." In physics, disorder is like throwing a bunch of random obstacles onto that dance floor—chairs, puddles, or uneven tiles. In normal physics, if you throw enough obstacles, the dancers get stuck in place, a state called "localization," and they stop dancing with each other. But what happens when you mix the unbalanced tilt (the skin effect) with the messy obstacles (disorder)? Does the tilt win, or does the mess win? This question is crucial because understanding how particles get "stuck" or "entangled" (a spooky quantum connection where particles share a fate) helps us build better quantum computers and understand how information flows in complex materials.

The Story of the Tilted, Messy Dance Floor

In this study, the researchers decided to play out this scenario using a famous model called the Hatano-Nelson model. Think of this model as a long, one-dimensional hallway with a floor that is slightly tilted to the left. The particles (fermions) are the dancers.

First, they looked at the hallway with no obstacles (no disorder). As expected, the tilt pushed all the dancers to the left wall. Because they were all crammed into one corner, they didn't have much "entanglement" with the rest of the hallway. It was a quiet, orderly pile-up.

Then, they started sprinkling in a little bit of disorder—just a few random obstacles. Surprisingly, this didn't just stop the dancers; it changed the way they connected. Instead of staying in a tight pile, the dancers began to form a "domain wall," a fuzzy, wavy line in the middle of the hallway where the density of dancers changed from full to empty. This fuzzy line grew larger as the hallway got bigger, but it grew slowly. The researchers found that the entanglement (the quantum connection) grew logarithmically with the size of the system. In everyday terms, if you doubled the length of the hallway, the entanglement didn't double; it increased by a tiny, steady amount, like a slow-growing vine.

But the story didn't end there. As the researchers kept adding more and more obstacles (increasing the disorder strength), something dramatic happened. The dancers got so confused by the mess that they stopped moving entirely. They got stuck in their individual spots, unable to interact with anyone else. The "fuzzy line" in the middle vanished, and the entanglement dropped back down to a low, constant level, independent of the hallway's size. This is called an "area-law" regime.

The moment between the slow-growing vine (logarithmic) and the stuck dancers (area-law) is where the magic happened. The researchers discovered a sharp phase transition. At a specific "critical point" of disorder (around a strength of 3.35), the system didn't just switch; it behaved in a unique, universal way. At this exact tipping point, the entanglement grew according to a power law, scaling with the system size to the power of roughly 0.5 (meaning it grew like the square root of the size). This suggests a special, critical state where the particles are neither fully stuck nor fully flowing.

What It's Not (and What It's Not Like)

The researchers were careful to point out what this new state is not. In many other quantum systems, when you see logarithmic growth in entanglement, it usually means the system is "conformally invariant"—a fancy way of saying it looks the same at every scale, like a fractal, and follows the strict rules of Conformal Field Theory (CFT).

However, the authors found that this disordered, tilted system does not follow those rules. They checked the "correlation functions" (how much one dancer knows about another far away) and the "mutual information" (how much two separate groups of dancers share). In a standard conformal system, these values would stay constant or follow a specific pattern. Here, they didn't. The "fuzzy line" created by the tilt and the disorder made the system behave differently than a standard fractal. The tilt broke the symmetry, making the system look different depending on which way you looked. So, while it looked like a conformal system in some ways, it was actually a new, distinct beast that didn't fit the old textbook definitions.

The Twist: The Loop vs. The Line

The researchers also tested what happens if the hallway is a loop (periodic boundaries) instead of having walls. Without disorder, the dancers circulate forever, and the entanglement grows logarithmically. When they added disorder, the dancers eventually got stuck, and the entanglement dropped to the "area-law" again. However, the transition was different here. In the loop, the system seemed to retain some "conformal-like" signatures, behaving more like the textbook fractal systems than the tilted hallway did. This suggests that the walls (boundaries) play a huge role in how the disorder and the tilt fight each other.

The Bottom Line

This paper suggests that when you mix the "skin effect" (particles piling up at an edge) with disorder (random obstacles), you don't just get a mess. You get a new kind of phase transition. The system moves from a state where entanglement grows slowly (logarithmically) to a state where it stops growing (area-law), passing through a critical point where it grows like a square root. Crucially, this new state breaks the usual rules of symmetry found in other quantum systems.

The authors simulated these scenarios on computers with systems up to 320 sites long, averaging over thousands of random disorder configurations to be sure. They found that the transition happens at a disorder strength of 3.35 (with a small margin of error), and the scaling exponent is approximately 0.5. While they didn't build a physical machine to prove this, their numerical evidence is strong, showing that the interplay between non-Hermitian skin effects and disorder creates a novel class of entanglement phase transitions that scientists have never seen before. It's a reminder that even in a messy, tilted world, quantum particles can find a new, strange order.

Technical Summary: Disorder-Induced Entanglement Phase Transitions in Non-Hermitian Systems with Skin Effects

Problem Statement
Non-Hermitian dynamics, prevalent in open quantum systems under continuous measurement and postselection, exhibits unique phenomena such as the non-Hermitian skin effect, where eigenstates accumulate at boundaries. Recent studies established that in disorder-free non-Hermitian systems with skin effects, the steady-state entanglement entropy follows an area law. However, the interplay between non-Hermitian skin effects and disorder remains poorly understood. Specifically, it is unclear how disorder modifies entanglement scaling in these systems, whether it induces entanglement phase transitions analogous to those in Hermitian systems (e.g., Anderson localization), and if such transitions exhibit distinct critical behaviors or conformal invariance.

Methodology
The authors investigate the dynamics of a half-filled many-body state of free fermions governed by the disordered Hatano-Nelson (HN) model. The system is described by a non-Hermitian Hamiltonian with asymmetric hopping ( $J_L \neq J_R$ ) and onsite disorder ( $m_i$ ) uniformly sampled from $[-W/2, W/2]$ .

Dynamics: The time-evolved state $|\psi(t)\rangle$ is defined via postselection (conditioned on no particle loss), effectively governed by $e^{-i\hat{H}t}$ . The initial state is a Néel state.
Numerical Approach: Since the Hamiltonian is quadratic and the initial state is a Slater determinant, the system remains a determinant state. The authors calculate the correlation matrix $D_{ij}(t)$ efficiently using a QR decomposition method at each time step to maintain numerical stability against exponential growth/decay inherent in non-Hermitian evolution.
Metrics: The study analyzes the von Neumann entanglement entropy ( $S_A$ ), subsystem entanglement entropy ( $S_l$ ), connected density-density correlation functions ( $C(l)$ ), and mutual information ( $I_{AB}$ ).
Boundary Conditions: The analysis is performed under both Open Boundary Conditions (OBCs) and Periodic Boundary Conditions (PBCs). Finite-size scaling is employed to determine critical points and exponents.

Key Contributions and Results

Discovery of Disorder-Induced Entanglement Phase Transitions:
- OBCs: In the pristine HN model ( $W=0$ $W = 0$ ), the system exhibits an area-law entanglement due to skin effects. As disorder strength $W$ $W$ increases, the system undergoes a transition:
  - Small $W$ : The area law evolves into a logarithmic scaling ( $S_{L/2} \propto \log L$ ). This regime arises because weak disorder cannot fully suppress the unidirectional shift, creating a broadened domain wall in the density profile.
  - Critical Point: At a critical disorder strength $W_c \approx 3.35$ (for $\gamma = -0.5$ ), the system exhibits a universal algebraic scaling $S_{L/2} \sim L^\beta$ with $\beta \approx 0.5$ .
  - Large $W$ : The system reenters an area-law regime ( $S_{L/2}$ independent of $L$ ), reminiscent of Anderson localization in Hermitian systems.
- PBCs: A similar log-to-area-law transition is observed. However, the critical behavior differs: while OBCs show algebraic scaling at the critical point, PBCs show the entanglement entropy converging to a logarithmic scaling even near the critical point, though the correlation exponents differ from the pure conformal case.
Characterization of Critical Exponents and Scaling:
- Using finite-size scaling of the form $S_{L/2}(W, L) = L^\beta F[(W - W_c)L^{1/\nu}]$ , the authors extract critical exponents: $\beta \approx 0.5$ and $\nu \approx 1.9$ .
- The transition point $W_c$ is identified via data collapse and is found to be distinct from the single-particle transition point for zero-energy states, suggesting the many-body nature of the transition.
Absence of Conformal Invariance in the Log-Law Regime (OBCs):
- Contrary to Hermitian critical systems where log-law scaling implies conformal invariance, the authors demonstrate that the log-law regime in the disordered HN model (OBCs) lacks conformal invariance.
- Evidence:
  - The subsystem entanglement entropy $S_l$ is zero for most subsystem sizes, showing a sharp peak only at the center, unlike the CFT prediction $S_l \propto \log[\sin(\pi l/L)]$ .
  - The connected correlation function $C(l)$ decays exponentially away from the critical point, rather than following the power-law decay expected in conformal systems.
  - The mutual information $I_{AB}$ between disjoint subsystems vanishes as system size increases, whereas it remains constant in conformal invariant systems.
- This absence is attributed to the non-reciprocal hopping causing a domain-wall structure in the density profile, where most of the system is in a pure state (fully occupied or empty).
Physical Mechanism:
- The transition is linked to the interplay between non-reciprocal hopping and disorder. Through a similarity transformation, the non-Hermitian Hamiltonian is mapped to a Hermitian one with modified hopping. The analysis suggests that at the critical point, single-particle eigenstates become "partially extended" in the region $x < x_n$ (where $x_n$ is the localization center), leading to the algebraic scaling of entanglement.
- The entanglement transition point aligns closely with the transition of the orthogonality index and the minimum of the Mean Inverse Participation Ratio (MIPR) of single-particle eigenstates.

Significance and Claims
The paper claims to identify a novel class of entanglement phase transitions driven by the interplay between non-Hermitian skin effects and disorder. The authors emphasize that:

Disorder can induce a transition from area-law to log-law and back to area-law scaling in non-Hermitian systems, a behavior distinct from the volume-law to area-law transition seen in Hermitian systems.
The log-law regime in these non-Hermitian systems does not possess conformal invariance, distinguishing it from standard critical phenomena in Hermitian systems or monitored Hermitian systems.
The results are not restricted to the HN model but are expected to apply to other non-Hermitian systems with skin effects (e.g., non-Hermitian Su-Schrieffer-Heeger model).
While postselection is required in the theoretical model, the authors note that recent work suggests measurements can induce skin effects in monitored systems without postselection, implying these transitions may be observable in trajectory-averaged entanglement entropy in experimental settings.

The study provides a comprehensive phase diagram and scaling analysis, establishing that non-Hermitian skin effects fundamentally alter the nature of disorder-induced entanglement transitions.

Technical Summary: Disorder-Induced Entanglement Phase Transitions in Non-Hermitian Systems with Skin Effects

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